H. N. Nath, U. Pyakurel, T. N. Dhamala, and S. Dempe ... · in pseudo-polynomial time. They also introduce the lex-maximum dynamic con-tra ow problem in which ow is maximized in given

ISSN 2512-3750

Fakultät für Mathematik und Informatik

PREPRINT 2018-12

H. N. Nath, U. Pyakurel, T. N. Dhamala, and S. Dempe

Dynamic Network Flow Location Models and Algorithms for Quickest Evacuation Planning

H. N. Nath, U. Pyakurel,

T. N. Dhamala, and S. Dempe

Dynamic Network Flow Location Models and Algorithms for Quickest Evacuation Planning

TU Bergakademie Freiberg

Fakultät für Mathematik und Informatik

Prüferstraße 9

09599 FREIBERG

http://tu-freiberg.de/fakult1

ISSN 1433 – 9307

Herausgeber: Dekan der Fakultät für Mathematik und Informatik

Herstellung: Medienzentrum der TU Bergakademie Freiberg

DYNAMIC NETWORK FLOW LOCATION MODELS AND

ALGORITHMS FOR QUICKEST EVACUATION PLANNING

HARI N. NATH1, URMILA PYAKUREL2, TANKA N. DHAMALA3 STEPHAN DEMPE4

Abstract. Dynamic network flow problems have wide applications in evacu-

ation planning. From a given subset of arcs in a directed network, choosingthe suitable arcs for facility location is very important in the optimization of

flows in emergency cases. Because of the decrease in the capacity of an arc by

placing a facility in it, there may be a reduction in the maximum flow or in-crease in the quickest time. In this work, we consider a problem of identifying

the optimal facility locations so that the increase in the quickest time is mini-

mum. Introducing the quickest FlowLoc problem, we give strongly polynomialtime algorithms to solve the single facility case. Realizing NP-hardness of the

multi-facility case, we develop a mixed integer programming formulation of it

and give a polynomial time heuristic for its solution. Because of the growingconcerns of arc reversals in evacuation planning, we introduce quickest Con-

traFlowLoc problem and present exact algorithms to solve single-facility case

and a heuristic to solve the multi-facility case, both of which have polynomialtime complexity. The solutions thus obtained here are practically important,

particularly in evacuation planning, to systematize traffic flow with facilityallocation minimizing evacuation time.

1. Introduction

The choice of locations for the facilities such as hospitals, warehouses, stores,fire-brigades, security offices, etc. plays an important role in normal as well as inemergency disastrous situations. As in the normal situations, the mathematicalmodels used to make location decisions in emergency situations are: (a) coveringmodels which locate the optimal locations to cover all demand points or maxi-mal number of demand points, (b) P -median models to determine P locations tominimize the average (or total) distance between demand points and facilities, (c)P -Center models to minimize the maximum distance between any demand pointand its nearest facility. For example, in Large Scale Emergency Medical ServiceFacility Location Model (LEMS) presented in Jia et al. [13], there is a use of theaforementioned models.

1991 Mathematics Subject Classification. 2010 Mathematics Subject Classification. Primary:90B10, 90C27, 68Q25; Secondary: 90B06, 90B20.

Key words and phrases. Network flow; facility location; evacuation planning; quickest flow;

contraflow.The research has been carried out under the AvH Research Group Linkage Program between

TU Bergakademie Freiberg, Germany and Central Department of Mathematics, TU Kathmandu,

Nepal. The first author acknowledges the support of UGC Nepal and TU Bergakademie Freiberg.The second author acknowledges the AvH for the George Foster Fellowship for Post-doctoral

Researchers.

1

2 HARI N. NATH1, URMILA PYAKUREL2, TANKA N. DHAMALA3 STEPHAN DEMPE4

Evacuation planning is an integral part of disaster management. Recently, thereis a growing trend of incorporating location decisions in evacuation planning. Thefollowing are some examples.

(i) Pick-up location models: An et al [2] formulate a model to determine theoptimal pick-up location, evacuee-to-facility assignment priorities, evacuationservice rates that minimizes the total expected system cost. In integrated busevacuation problem, Goerigk et al. [10] choose pick-up locations to minimizethe maximum travel time over all buses. Kulshrestha et al. [17] use robustoptimization to locate pick-up locations when number of evacuees is uncertain.

(ii) Rescue Transfer Location Models: An et al. [2] formulate a model to locaterescue transfer locations, where rescue team departs from the rescue center,rides a vehicle to rescue transfer center, and walks to each rescuee group toprovide aid with an objective to minimize the total expected travel cost.

(iii) Shelter Location Models: Sherali et al. [36] formulate a location allocationmodel to minimize the total vehicle hours, and a discrete median locationmodel to locate shelters for evacuation, while Kongsomsaksakul et al. [15]use bi-level programming approach to determine shelter locations, in whichthe upper level determines the shelter locations to minimize the total evac-uation time and the lower level is formulated as combined trip distributionand assignment problem. Ng et al. [19] also use the same approach in whichthe lower level is deterministic user equilibrium model as described by Sheffi[37]. In integrated bus evacuation problem, Goerigk et al. [10] choose sheltersto minimize the maximum travel time over all buses while in comprehensiveevacuation planning, Goerigk et al. [9] formulate a multiple commodity, multi-criteria problem to minimize total evacuation time, risk exposure of evacuees,and number of shelters that are used.

(iv) Flow Location (FlowLoc) Models: Optimizing traffic flow is a very importantaspect of evacuation planning. The common methods to optimize traffic floware traffic simulation, models based on fluid dynamics, control theory, vari-ational inequalities, and network flow. Since simulation does not explicitlyallow for optimization, and models based on differential equations are not ca-pable of handling large networks, network flow approach has been the mostappropriate way of modeling traffic flow (Kohler and Skutella [16]). For de-tails on network flow approach for evacuation planning problems, we refer toDhamala et al. [5]. Rupp [33], Hamacher et al. [11], Heller and Hamacher[12] combine location decisions with flow decisions in a network flow problemobserving that the placement of a facility on an arc of a network may resultinto a reduction in the maximum flow value. Given a set of facilities andset of arcs on which facilities are to be placed, their approach is to find anallocation of the facilities to the arcs so that the reduction in the maximumflow is minimum.

The main aim of optimizing traffic flow in an emergency evacuation process is tomaximize flow and/or to minimize the evacuation time using the road network. Insuch situations, people are discouraged to go towards risk areas from safer places.As a result the road segments heading towards the safe areas become overly con-gested and those heading towards the risk areas become empty. To maximize theflow and to minimize the evacuation time, in such situations, converting a two-wayroad segment to one-way in an appropriate direction becomes advantageous. This

DYNAMIC NETWORK FLOW LOCATION MODELS AND ALGORITHMS 3

is known as contraflow configuration, which reverses the direction of the traffic onempty road segments towards the sinks so that the capacity of the road segment isincreased. Contraflow configuration not only increases flow value but also reducesthe traffic jam and makes the traffic smooth, [21]. But to identify appropriatedirection of the arcs of a network to maximize the flow is a difficult optimizationproblem, known as a contraflow problem. Different heuristic techniques to solvethe contraflow problem in which at least 40% evacuation time can be reduced byreverting at most 30% arcs can be found in Kim et al. [14].

Apart from heuristic techniques, recent research also focuses on analytical tech-niques to find exact solution of contraflow problem after Rebennack et al. [32] in-troduced algorithms to solve the single-source single-sink maximum contraflow andquickest contraflow problems optimally in polynomial time. The earliest arrivaland the maximum contraflow problems are solved with the temporally repeatedflow solutions in Dhamala and Pyakurel [4, 22] with discrete time setting. Thesolution procedures for such problems in continuous time setting are described inPyakurel and Dhamala [24]. In Pyakurel and Dhamala [23], authors design algo-rithms to solve the earliest arrival contraflow on single-source single-sink networkin pseudo-polynomial time. They also introduce the lex-maximum dynamic con-traflow problem in which flow is maximized in given priority ordering and constructsolution algorithms with polynomial time complexity. Algorithms to these prob-lems in continuous time by using nice property of natural transformation can befound in Pyakurel and Dhamala [24, 26]. With given supplies and demands, theearliest arrival transshipment contraflow problem is modeled in discrete time andsolved it on multi-source network with polynomial time algorithm in Pyakurel andDhamala [25]. With zero transit time on each arc, the problem is also solved onmulti-sink network in polynomial time complexity. For the multi terminal network,they present approximation algorithms to solve the earliest arrival transshipmentcontraflow problem. The discrete solutions are extended into continuous time inPyakurel and Dhamala [26]; Pyakurel et al. [27]. The maximum dynamic andearliest arrival contraflow problems are generalized in Pyakurel et al. [31].

Moreover, the first temporally repeated flow algorithm to solve the quickestcontraflow problem has been presented by Pyakurel et al. [28]. Considering a caseof Kathmandu road network, the comparison of the quickest time before and aftercontraflow configuration show that a significant decrease in the quickest time can beattributed to contraflow configuration and the decrease in quickest time increaseswith the number of evacuee-vehicles. They also present an approximate algorithmto solve the quickest contraflow problem with load dependent transit time on eacharc.

The analytical techniques discussed above use arc-based formulation of networkflow problems. Recently path-based formulation of the similar problem with ab-stract flow on abstract networks is also gaining attention. Pyakurel et al. [27]introduce contraflow technique in abstract networks, present algorithms to solvemaximum static and maximum dynamic contraflow problems with continuous timesetting and realize that if the minimum dynamic cut capacities on two terminal net-work are symmetric, then the flow value can be increased up to double with partialcontraflow reconfiguration. The models and algorithms for the abstract contraflowproblems with discrete time setting have been investigated in Dhungana et al. [7].With a view to save unused capacities of arcs during evacuation process, Pyakurel


et al. [29, 30] investigate the partial contraflow problem and present algorithms tosolve various problems related to abstract flow.

Motivated by the work of Hamacher et al. [11], our main focus in this paper is tointroduce flow location models and develop efficient solution procedures to identifythe allocation of facilities on arcs, with and without contraflow configuration, sothat the increase in the quickest time is minimum. To facilitate the evacuationprocess, placing facilities on the road segments obstruct traffic flow resulting in theincrease of the evacuation time. From a given set of road segments for the facilitiesto be placed, our approach is to choose those which have minimum impact onthe increase in the transportation time. This is important in evacuation planning,particularly, when the given number of evacuees are to be transferred to the safedestinations as quickly as possible.

The paper is organized as follows. The basic terminology, notations and flowmodels necessary to the paper is considered in Section 2. Section 3 investigates thequickest FlowLoc problem and presents strongly polynomial algorithms to solvesingle facility cases and polynomial heuristic to solve multiple facility case. InSection 4, we combine contraflow with location decision on arcs and Section 5concludes the paper with further research directions.

2. Basic concepts

In this section, we give some basic ideas used in this paper in an attempt tomake it self-contained. We represent a transportation network by a directed graphin which the intersections of roads (or some other points on a road, if needed)denote the nodes and the road segments between any two nodes represent arcs. Thedirection of the traffic flow in a road segment is the direction of the correspondingarc. Something that moves from one node to the other via arcs is known as a flow.

We represent a directed network (also known as evacuation network in this paper)with the set of nodes V , set of arcs A, capacity b : A → R≥0, travel time τ : A →R≥0, the source node s ∈ V and the sink node d ∈ V by N = (V,A, b, τ, s, d). Thecapacity of an arc limits the flow rate on the arc and travel time represents thetime the flow takes to travel on the arc. We denote the number of nodes |V | by n,the number of arcs |A| by m, the set of incoming arcs to the node i by Ain

i and theset of arcs going out of it by Aout

i i.e.

Aini = {e ∈ A : e = (j, i) for some j ∈ V }

Aouti = {e ∈ A : e = (i, j) for some j ∈ V }

2.1. Static and dynamic flows. Let x : A→ R≥0 be a function of non-negativevalues, where x(e) or xe is considered as the flow value on e ∈ A. For each i ∈ V ,we denote the excess of x at i ∈ V by

(2.1) excx(i) =∑e∈Ain

i

x(e)−∑

e∈Aouti

x(e)

which is the difference of the flow value entering i and that going out from i.

Definition 2.1. For two distinct nodes s, d ∈ V , x is called a static s-d flow if

(2.2) excx(i) = 0, ∀i ∈ V \ {s, d}.The static flow x is called feasible if

(2.3) 0 ≤ x(e) ≤ b(e), ∀e ∈ A.


The value of x is defined as:

(2.4) val(x) = excx(d).

If excx(i) = 0 for all i ∈ V , then x is called a circulation.

The famous maximum (static) flow problem aims at finding a feasible static flowx that maximizes val(x). For details, we refer to Ahuja et al. [1] and Dhamala etal. [5].

The above-discussed formulation is the arc-flow formulation of a static flow. Analternative to this approach is the path and cycle flow formulation. Let Γ be thecollection of all the s-d paths and C be the collection of cycles of the network. Letf(γ) and f(C) be the flows in γ ∈ Γ and C ∈ C, then the arc flow is

(2.5) x(e) =∑γ∈Γ

δe(γ)f(γ) +∑C∈C

δe(C)f(C)

where δe(γ) = 1 if e ∈ γ, and zero otherwise. Similarly, δe(C) = 1 if e ∈ C, andzero otherwise. The relation (2.5) determines x uniquely if path and cycle flowsare given. Conversely, given an arc flow x, we can find path (s-d path) and cycleflow f (not necessarily unique) such that (2.5) is satisfied. This is called the flowdecomposition of x into path and cycle flows. For more details, we refer to Ahujaet al. [1].

A very important concept in flow optimizations is the residual network. Wedenote the static residual network of N = (V,A, b, τ, s, d) with respect to the flowx by N(x). N(x) has the same vertex set V and an arc set A(x) = AF (x)∪AB(x)where AF (x) = {(i, j) | x(i, j) < b(i, j)} and AB(x) = {(j, i) | x(i, j) > 0}. For(j, i) ∈ AB(x), τ(j, i) = −τ(i, j). In the residual network N(x), we define theresidual capacity bx : A(x)→ R by

bx(i, j) =

{b(i, j)− x(i, j) if (i, j) ∈ AF (x)

x(j, i) if (i, j) ∈ AB(x)

The relation of the static flow with the residual network is that whenever thereexists a path from the source s to the sink d in the residual network, the value ofthe flow can be increased.

A dynamic flow Φ with time horizon T consists of Lebesgue-integrable functionsΦe : [0, T ) → R≥0 for each arc e ∈ A such that Φe(θ) = 0 for θ ≥ T − τ(e). Φe(θ)can be realized as the rate of flow entering e at time θ. The flow entering the tail iof the arc e = (i, j) at time θ reach the head j of e at time θ + τe. For each i ∈ V ,we define the excess of node i induced by Φ at time θ as:

(2.6) exΦ(i, θ) =∑e∈Ain

i

∫ θ−τ(e)

0

Φe(σ)dσ −∑

e∈Aouti

∫ θ

0

Φe(σ)dσ

which is the net amount of flow that enters node i up to time θ.

Definition 2.2. A feasible dynamic s-d flow (s, d ∈ V and s 6= d) satisfies:

(2.7) exΦ(i, θ) ≥ 0 ∀θ ∈ [0, T ), ∀i ∈ V \ {s}

(2.8) exΦ(i, T ) = 0, ∀i ∈ V \ {s, d}


and

(2.9) 0 ≤ Φe(θ) ≤ b(e), ∀e ∈ A, θ ∈ [0, T ).

The value of the dynamic flow Φ at time θ is

valθ(Φ) = exΦ(d, θ)

and the total value of the dynamic flow Φ is:

val(Φ) = valT (Φ) = exΦ(d, T )

For more details, we refer to Skutella [38].In the course of designing efficient algorithms related to a dynamic flow, a dy-

namic flow is represented as what is known as temporally repeated flow. Given afeasible static flow x and a time horizon T , a flow decomposition on x gives a setof paths Γ with flow f(γ) for each γ ∈ Γ. Flow is sent along γ at a constant ratef(γ) from time 0 to max{T − τ(γ), 0}, where τ(γ) =

∑e∈γ τ(e) is the travel time

on path γ. In this way, the dynamic flow is obtained as described in the followingequation

(2.10) Φe(θ) =∑

γ∈Γe(θ)

f(γ), ∀e = (i, j) ∈ A, θ ∈ [0, T )

where Γe(θ) = {γ ∈ Γ|e ∈ P and τ(γs, i) ≤ θ and τ(γj , d) < T − θ}. Here, τ(γs, i)is the sum of times of the arcs from node s to node i on path γ and τ(γj , d) is thesum of times of the arcs from node j to node d on path γ.

Given a time horizon T , the maximum dynamic problem seeks to find a dynamicflow Φ which maximizes valT (Φ). Using temporally repeated flows, Ford and Fulk-erson [8] showed that finding a maximum dynamic flow is equivalent to finding aminimum circulation x that minimizes

∑e∈A τ(e)x(e) − T · val(x) adding an arc

(d, s), to the network, with infinite capacity and −T cost(time). From the flowdecomposition of x, one can find the dynamic maximum flow in the temporallyrepeated form using equation (2.10).

Given a time horizon T , we say that a dynamic flow has the earliest arrivalproperty if as much flow as possible arrives at the sink at each time θ < T and thecorresponding flow is called earliest arrival flow. Every earliest arrival flow is also amaximum dynamic flow but the converse is not true in general (Ruzika et al. [34]).

A problem closely related with the maximum dynamic flow problem is the quick-est flow problem which seeks to find dynamic flow with minimum time horizon T ∗

needed to send a given amount of flow F from the source s to the sink d.Using the idea of finding a dynamic flow with the temporal repetition of the

static flow, the following mathematical programming formulation of the problem isuseful in designing algorithms to find a quickest flow.

Theorem 2.3 (Lin and Jaillet [18]). The quickest flow problem can be formulatedas the following fractional programming problem:


minF +

∑e∈A τ(e)x(e)

v(2.11)

s.t.∑

e∈Aouti

x(e)−∑e∈Ain

i

x(e) =

v if i = s

−v if i = d

0 otherwise

(2.12)

0 ≤ x(e) ≤ b(e), ∀e ∈ A.(2.13)

Observing that if v is fixed in the above formulation, it becomes a min-cost flowproblem with supply at s and demand at d both equal to v, they realize that thequickest flow problem is a parametric min-cost flow problem with respect to v.Deriving optimality conditions on this basis, they design a cost-scaling algorithmto solve the quickest flow problem with polynomial time complexity O(n3 log(nC))of a min-cost flow problem, where C is the maximum arc cost. Stepping on theirapproach Saho and Sigeno [35] design a cancel-and-tighten algorithm which runs instrongly polynomial time, O(nm2 log2 n), to solve the quickest flow problem.

3. Combining location decisions with the quickest flow

If a facility is placed on an arc of a network, it reduces the capacity of corre-sponding arc affecting the decisions related to flow. Hamacher et al. [11] model suchproblems so that there is the least reduction in the maximum flow value because ofthe placement of the facility.

Definition 3.1 (Maximum static and dynamic FlowLoc). Let N = (V,A, b, τ, s, d)be a network with set of all feasible locations L ⊆ A, set of all facilities P , thesize of the facilities r : P → N and the number of facilities that can be placedon the possible locations ν : L → N. The maximum static FlowLoc problem asksfor an allocation π : P → L, such that the static s-d flow value in the networkNπ = (V,A, τ, bπ, s, d) is maximized where the capacity function bπ is defined asbπ(e) = b(e) − max{r(p) : p ∈ P and π(p) = e}. In the above setting, if thedynamic s-d flow value is to be maximized, it is called a maximum dynamic FlowLocproblem. If |P | = 1, the corresponding problem is called a single facility maximumstatic (dynamic) FlowLoc problem and if |P | = q > 1, then it is referred to as astatic (dynamic) multi facility FlowLoc or a q-FlowLoc problem.

Because of the reduction in the arc capacity, placing a facility on an arc of anetwork may result into the increase in the time to transfer a given amount of flowfrom the source the sink. In the following definition, we consider the problem oflocating the facilities so that the increase in the quickest time is minimum.

Definition 3.2 (Quickest FlowLoc). Given a network N = (V,A, b, τ, s, d), asupply F at s, set of feasible locations L ⊆ A, the set of all facilities P , thesize of the facilities r : P → N, the number of facilities that can be placedon the possible locations ν : L → N, the quickest FlowLoc problem seeks foran allocation π : P → L of the facilities to the edges, such that the quickesttime to transport F from s to d on the network Nπ = (V,A, bπ, τ, s, d), wherebπ(e) = b(e)−max{r(p) : p ∈ P and π(p) = e}, is minimized.

Remark 1. In our considerations, the set of feasible locations L and the set offacilities P are to be given in such a way that


1

2

3

4

4,13,1

4,33,3

3,43,4

3,11,1

1,1 1,1

Figure 1. Evacuation network N with arc labels (capacity, travel time)

(i) |P | ≤∑e∈L ν(e) and

(ii) r(p) ≤ min{b(e) : e ∈ L}, ∀p ∈ P

The following example gives the comparison of location decisions on arcs underdifferent flow decisions.

Example 1. Consider the evacuation network depicted in Figure 1. The pair ofnumbers on each arc represents capacity and travel time related to the arc. LetP = {p}, r(p) = 1, L = {(2, 3), (2, 4)}, i.e. a facility p of size r(p) = 1 is to be placedon one of the arcs in L. If the facility is placed on (2,3), the maximum static flowvalue is 7 (4 along the path 1 − 2 − 4 and 3 along the path 1 − 3 − 4), while if itis placed on (2,4), the maximum static flow value is 6 (3 along the path 1− 2 − 4and 1− 3− 4 each). Hence, π(p) = (2, 3) is the maximum static FlowLoc decision,which does not consider the time factor associated with arcs.

Table 1 shows that maximum dynamic FlowLoc decisions depend on the timehorizon T . In the continuous time setting, when T = 4, if no facility is placed,a flow of value 1 can reach the sink using only the path 1 − 2 − 3 − 4. So if thefacility is placed on the arc (2, 3), this path gets obstructed and no flow can reachthe sink, so (2, 4) is the optimal location in this case. When T = 5, if the facilityis placed on (2, 3), the flow of value 4 can reach the sink via path 1− 2− 4, whilethe flow of value 5 can reach the sink if we place the facility on (2, 4) using thepath 1 − 2 − 3 − 4 with flow value 1 twice and path 1 − 2 − 4 with flow value 3once. Table 2 shows that the quickest FlowLoc decisions depend on the flow valueF to be transferred from s to d. The calculation of the quickest time is done usingcost scaling algorithm by Lin and Jaillet [18] (see Section 3.1.1). As expected, themaximum dynamic FlowLoc and quickest FlowLoc decisions are related with eachother in some sense.

3.1. Single facility quickest FlowLoc. In this section, we design efficient algo-rithms to solve a single facility (|P | = 1) quickest FlowLoc problem. To set up abackground, we discuss the single facility static and dynamic FlowLoc problems.

To solve the single facility static FlowLoc problem, Hamacher et al. [11] givethree algorithms. The main idea is to iterate over the arcs in L, place the facility,calculate the maximum flow value and finally choose the arc which gives the greatestmaximum flow value to place the facility. With preflow push algorithm to perform


Table 1. Maximum dynamic FlowLoc decisions (cf. Figure 1)

Tmaximum dynamic flow value

Location decisionwhen facility is placed on(2,3) (2,4)

4 0 1 (2,4)5 4 5 (2,4)6 11 11 (2,4) or (2,3)7 18 17 (2,3)

Table 2. Quickest FlowLoc decisions (cf. Figure 1)

Fquickest time

Location decisionwhen facility is placed on(2,3) (2,4)

1 4.25 4 (2,4)5 5.14 5 (2,4)11 6 6 (2,4) or (2,3)21 7.43 7.67 (2,3)

maximum flow calculations, the time complexity of such an algorithm, as realizedin [11], is O(|L|n3).

Observation 1. Using the algorithm given by Orlin [20] to find the maximumflow, the time complexity of single facility static FlowLoc problem can be reducedto O(|L|mn).

In case of a single facility dynamic FlowLoc, a similar idea can be used calculatinga maximum dynamic flow with the help of temporally repeated static flow. Givena time horizon T , to calculate the temporally repeated static flow corresponding tothe maximum dynamic flow, one can take τ as cost, add an arc (d, s) in the networkwith infinite capacity and cost −T , and calculate minimum cost circulation in themodified network. As suggested in [11], we present Algorithm 1 to solve the singlefacility dynamic FlowLoc.

Observation 2. Using the dual network simplex algorithm presented in Armstrongand Jin [3] for solving the minimum cost flow problem, Algorithm 1 solves thesingle facility dynamic FlowLoc problem in O(|L|mn(m+ n log n) log n) time. In aseries parallel graph (Ruzika et al. [32]), the maximum dynamic flow problem withtime horizon T can be solved in O(mn + m logm) time, using greedy approach,by sending the flow iteratively through the s-d path with the minimum time andremoving the saturated arc, considering only the paths with time not exceedingT . Thus, in a series parallel graph, the single facility maximum dynamic FlowLocproblem can be solved in O(|L|(mn + m logm)) time. Moreover, in such a graph,a maximum flow has also the earliest arrival property which requires the flow tobe maximized at each period of time. Thus, maximum dynamic FlowLoc decisionsunder earliest arrival flow in a series parallel graph can also be solved with thesame time complexity, although finding earliest arrival flow in a general graph hasa pseudopolynomial time complexity.


Algorithm 1: Single facility maximum dynamic FlowLoc

Input : Directed network N = (V,A, b, τ, s, d), the set of possible locationsL, time horizon T , size r of the facility

Output: Location loc of the facility, the static flow x corresponding to themaximum dynamic flow

1 Add an arc (d, s) with infinite capacity and cost −T to N and consider τ ascost to obtain a network N c

2 curr max flow = −1

3 for l ∈ L do4 b(l) = b(l)− r5 x′ = min-cost circulation in N c

6 new max flow = −cost of x′

7 if new max flow > curr max flow then8 curr max flow = new max flow

9 loc = l

10 x = restriction of x′ to N

11 end

12 b(l) = b(l) + r

13 end

14 return loc, x

Before proceeding further, we summarize two algorithms to solve the quickestflow problem, which are central in designing algorithms to solve quickest FlowLocproblems.

3.1.1. Cost scaling algorithm (Lin and Jaillet, [18]). Given N = (V,A, b, τ, s, d) asan evacuation network with a supply F at s, let x be a static flow with value v.Node potentials ρ are introduced and the reduced cost ce = ρ(j) − ρ(i) + τ(e) iscalculated for each arc e = (i, j) ∈ N(x), the residual network corresponding to x.When ce > −ε,∀e ∈ N(x), the obtained flow x is called ε-optimal. The algorithmis briefly described in the following steps.

(1) Initialize: ρ(u) = 0, ∀ u ∈ V , x(e) = 0,∀e ∈ A, and ε = C = maxe∈A{τ(e)}.(2) Refine: The 2ε-optimal flow is modified to an ε-optimal one by assigning

the flow in the arcs of N with ce < 0 to their capacity, assigning zero flowin the arcs with ce > 0, then pushing flows from nodes with excess flowthrough the arcs in the residual network N(x), relabeling their potential ifrequired.

(3) Reduce Gap: Set extra flow at s and push the admissible flow ultimatelyto d with arcs in N(x), and relabel the potential of nodes if required toreduce the gap between T = [F +

∑e∈A τ(e)x(e)]/v and ρ(s)− ρ(d) by at

least 7nε.

After Step 3, ε is scaled by 1/2, and Steps 2 and 3 are repeated unless it becomesless than 1/8n.

(4) Saturate: If T , obtained from the above-mentioned scaling phases, is morethan the time (cost) in a shortest simple path from s to d in the residualnetwork N(x), the flow is saturated by sending maximum flow from s to d


in a subnetwork N ′, formed by only those arcs which are on some shortestpath from s to d in N(x).

The time complexity of the cost-scaling algorithm is O(n3 log(nC)). For details, werefer to [18].

3.1.2. Cancel-and-tighten algorithm (Saho and Sigeno, [35]). The main idea of thealgorithm is to modify the cost scaling algorithm replacing Step 2 with Cancel andTighten steps.

• Cancel: Find a cycle in N(x) with only admissible arcs (an arc e ∈ N(x)is admissible if its reduced cost ce < 0) and push a flow equal to minimumresidual capacity of its arcs. Repeat the process until there remains no suchcycle.• Tighten: For each node u, compute the maximum length h(u) from nodes

with no entering admissible arc. Replace ρ(u) by ρ(u) + εnh(u) and reduce

ε to (1− 1n )ε.

Cancel Step and Tighten Steps are repeated iteratively until ε reduces to ε/2.Then Reduce Gap step reduces the gap between T = [F +

∑e∈A τ(e)x(e)]/v and

ρ(s)− ρ(d) by at least (3n+ 1)ε in this case.The above-mentioned steps are performed until ε becomes smaller than 1/4n,

and finally Saturate step is performed as in cost scaling algorithm. The complexityof this algorithm is O(nm2 log2 n).

Now, we construct two algorithms for the single facility quickest FlowLoc problem.Algorithm 2 iterates through all possible locations l ∈ L, determines the quickesttime if location l hosts the facility and finds the optimal location for the singlefacility by comparing all those quickest times. It also records the quickest flow, andthe quickest time after placing the facility.

Algorithm 2: Single facility quickest FlowLoc I

Input : Directed network N = (V,A, b, τ, s, d), the set of possible locationsL, supply F at s, size r of the facility

Output: Location loc of the facility, the corresponding static flow x, thecorresponding quickest time T

1 T =∞2 for l ∈ L do3 b(l) = b(l)− r4 new quickest time = the quickest time in the modified network

5 if new quickest time < T then6 T = new quickest time

7 loc = l

8 x = static flow corresponding to the quickest flow

9 end

10 b(l) = b(l) + r

11 end

12 return loc, x, T

Algorithm 2 performs the quickest flow computations |L| times. If we perform asingle quickest flow computation before going through Algorithm 2, and find that


an arc in L has residual capacity enough to accommodate the given facility, we canget rid of |L|−1 quickest flow computations in Algorithm 2. Algorithm 3 addressesthis issue.

Algorithm 3: Single facility quickest FlowLoc II

Input : Directed network N = (V,A, b, τ, s, d), the set of possible locationsL, supply F at s, size r of the facility

Output: Location loc of the facility, the corresponding static flow x, andthe quickest time

1 x = static flow corresponding to the quickest flow in the network N

2 T = the corresponding quickest time

3 e = arg max{b(l)− x(l)|l ∈ L}4 if b(e)− x(e) ≥ r then5 loc = e

6 b(e) = b(e)− r7 else8 Algorithm 2

9 end

10 return location loc, x, T

Theorem 3.3. The single facility quickest FlowLoc problem can be solved in stronglypolynomial time.

Proof. In the worst case, Algorithm 2 or Algorithm 3 iterates the quickest flow com-putations |L| times. The cancel-and-tighten algorithm by Saho and Shigeno [35](seeSection 3.1.2), the quickest flow problem can be solved in O(nm2 log2 n) time.Hence, the single facility quickest FlowLoc problem can be solved inO(|L|nm2 log2 n)time.

�

Example 2. To illustrate the working of Algorithm 2 and Algorithm 3, we considerthe network depicted in Figure 1 with L = {(1, 2), (1, 3), (2, 3)}, r = 1, F = 11. InAlgorithm 2, we take curr quickest time = ∞. In the first iteration, we take l =(1, 2), reduce its capacity 4 by 1 and calculate the quickest time which is 6.33 <∞.So loc = (1, 2) and the capacity of (1, 2) is retained to 4. In this way, after the thirditeration, we get loc = (1, 3) after three quickest flow calculations. However, inAlgorithm 3, we calculate the static flow x associated with the quickest flow in thebeginning (see the first table in Example 3) and see that b(1, 3) − x(1, 3) = 1 ≥ rso that loc = (1, 3).

Observation 3. Let |P | > 1 and ν(e) ≥ |P |, ∀e ∈ L, p∗ = arg max{r(p) : p ∈ P}.If e∗ ∈ L is the single facility location, taken p∗ as the single facility, then π(p) =e∗, ∀p ∈ P .

3.2. Multi-facility quickest FlowLoc. Now we consider the quickest FlowLocproblem for |P | > 1. The idea of the single facility case (i.e. iterating over allthe possibilities to locate facilities) can be carried over to the multiple facility casealso. As described in Hamacher et. al. [11], this does not lead to a polynomial


algorithm even in case of static FlowLoc, the exception being the case mentionedin Observation 3. The following result is crucial in this regard.

Theorem 3.4 (Hamacher et al.,[11]). There is no polynomial time α-approximationalgorithm for the multi-facility maximum static FlowLoc problem with a finite con-stant α unless P = NP .

As we have seen that the multi-facility static maximum FlowLoc problem is NP -hard, we realize the hardness of the multi-facility quickest FlowLoc problem in thefollowing lemma.

Lemma 3.5. For F > 0, if τe = 0 ∀e ∈ A, the quickest flow problem is equivalentto the maximum static flow problem.

Proof. The maximum static flow problem can be stated as:

max v(3.1)

∑e∈Aout

w

xe −∑e∈Ain

w

xe =

v if w = s

−v if w = d

0 if w /∈ {s, d}(3.2)

0 ≤ xe ≤ be ∀e ∈ A(3.3)

The objective function (3.1) replaced byF+

∑e∈A xeτev with the same constraints

gives the quickest flow problem because of Theorem 2.3. If τe = 0 ∀e ∈ A, thequickest flow problem reduces to minimize F/v subject to the constraints (3.2) and(3.3). But for a fixed F > 0, minimizers of F/v will maximize v. �

Theorem 3.6. There is no polynomial time α-approximation algorithm to solvethe multifacility quickest FlowLoc problem unless P = NP .

Proof. Suppose that there is a polynomial time α-approximation algorithm to solvethe multifacility quickest FlowLoc problem for α < ∞. According to Lemma 3.5,the maximum static flow problem is a special case of the quickest flow problem.This implies that there exists such an algorithm for multi-facility static FlowLocproblem which contradicts Theorem 3.4. �

Because of Theorem 3.6, we design a polynomial time heuristic to solve the multi-facility quickest FlowLoc problem presenting its mixed integer programming formu-lation. The mathematical programming formulation of the multi-facility quickestFlowLoc problem, based on Theorem 2.3, is as follows.


minF +

∑e∈A τexe

v(3.4)

∑e∈Aout

i

xe −∑e∈Ain

i

xe =

v if i = s

−v if i = d

0 if i ∈ V \ {s, d}(3.5)

xe + yeprp ≤ be, ∀e ∈ L, p ∈ P(3.6)

0 ≤ xe ≤ be, ∀e ∈ A(3.7) ∑e∈L

yep = 1, ∀p ∈ P(3.8) ∑p∈P

yep ≤ νe, ∀e ∈ L(3.9)

yep ∈ {0, 1}, ∀e ∈ L(3.10)

The variables and constants used in the model are described as follows.Variablesxe = static flow corresponding to the quickest flow in e ∈ A

yep =

{1 if the facility p is placed on e ∈ L0 if the facility p is not placed on e ∈ L

Constantsrp = r(p), the size of the facility pbe = b(e), the capacity of e ∈ Aνe = ν(e) = the number of facilities that can be placed on e ∈ LConstraints (3.5) and (3.7) are conditions for a static flow. Constraints (3.6)

reduce the capacity of e by rp if the facility p is placed on e. Constraints (3.8) statethat each facility has to be placed in exactly one arc, and constraints (3.9) boundthe number of facilities on an arc by the admissible number of facilities on it.

The objective function of the above problem is not linear. To make it linear, weput 1/v = θ, and xeθ = ξe. As a result the problem becomes

min Fθ +∑e∈A

τeξe(3.11)

∑e∈Aout

i

ξe −∑e∈Ain

i

ξe =

1 if i = s

−1 if i = d

0 if i ∈ V \ {s, d}(3.12)

ξe + θyeprp ≤ beθ, ∀e ∈ L, p ∈ P(3.13)

0 ≤ ξe ≤ beθ, ∀e ∈ A(3.14) ∑e∈L

yep = 1, ∀p ∈ P(3.15) ∑p∈P

yep ≤ νe, ∀e ∈ L(3.16)

yep ∈ {0, 1}, ∀e ∈ L, p ∈ P(3.17)

However, the set of constraints (3.13) are not linear. If one wants to use a linearmixed integer programming solver to solve the model, one can linearize them using


the idea given in Torres [39], replacing (3.13) with the following constraints ∀e ∈L, p ∈ P

ξe + ζeprp ≤ beθ(3.18)

ζep ≤ Myep(3.19)

ζep ≤ θ(3.20)

ζep ≥ θ − (1− yep)M(3.21)

ζep ≥ 0(3.22)

where M is an upper bound on the value of θ which can be taken 1 if there is atleast one path from the source to sink with positive integral capacities and F isalso a positive integer, because v is at least 1 in such cases.

Now, we construct a polynomial time heuristic, in Algorithm 4, to solve the prob-lem that works well in practice. First of all, the facilities are sorted in decreasingorder of their sizes. Then the quickest flow calculation is done (polynomial timealgorithms for such calculations exist, see Section 3.1.1 and 3.1.2) and the residualcapacities of the arcs in L are calculated. Then, first ν(e) facilities are placed onthe arc e with the largest residual capacity, and e is removed from L. The process isrepeated until all the facilities are allocated to some or all arcs in L. If the residualcapacity of e is less than the size of the largest facility hosted by it, the quickestflow is recalculated in Line 15.

Algorithm 4: Multi facility quickest FlowLoc heuristic

Input : Directed network N = (V,A, b, τ, s, d), the set of possible locationsL, supply F at the source s, set of facilities P with size r : P → N

Output: Allocation π : P → L, the quickest time T after allocation1 sort facilities according to their size r(p1) ≥ r(p2) ≥ · · · ≥ r(pq)2 x = static flow corresponding to the quickest flow


4 i = 1

5 while i ≤ q do6 e = arg max{b(l)− x(l)|l ∈ L}7 for j = 1 to j = ν(e) do8 if i+ j − 1 ≤ q then9 π(pi+j−1) = e

10 end

11 end

12 L = L \ {e}13 b(e) = b(e)− r(pi)14 if b(e)− x(e) + r(pi) < r(pi) then15 x = static flow corresponding to the quickest flow with modified b


17 end

18 i = i+ ν(e)

19 end

20 return π, x, T


Example 3. To illustrate Algorithm 4, we consider the network given in Fig-ure 1. Let L = {(2, 1), (2, 4), (1, 3), (3, 4)} with ν(2, 1) = 1, ν(2, 4) = 2, ν(1, 3) =1, ν(3, 4) = 3 and P = {f1, f2, f3, f4} with r(f1) = 1, r(f2) = 3, r(f3) = 2, r(f4) = 1.

First of all, we order the facilities in the decreasing order of their size, i.e. p1 =f2, p2 = f3, p3 = f1, p4 = f4 so that r(p1) = 3, r(p2) = 2, r(p3) = 1, r(p4) = 1.

The static flow corresponding to the quickest flow is given in the following table.

arc (1, 2) (1, 3) (2, 1) (2, 4) (2, 3) (3, 1) (3, 2) (3, 4) (4, 2) (4, 3)b 4 3 3 4 1 3 1 3 3 1x 4 2 0 3 1 0 0 3 0 0b(l)− x(l)|l ∈ L 1 3 1 0

T = 6q = 4i = 1 < q

First iteration:e = max{b(e)− x(e)|e ∈ L} = (2, 1)j = 1i+ j − 1 = 1 + 1− 1 = 1π(p1) = (2, 1)L = {(2, 4), (1, 3), (3, 4)}b(e) = 3− 3 = 0b(e)− x(e) + r(pi) = 0− 0 + 3 ≮ r(p1)i = 1 + 1 = 2 < q

Second iteration:e = max{b(e)− x(e)|e ∈ L} = (2, 4) (We may take e = (1, 3) also.)j = 1, 2i+ j − 1 = 2 + 1− 1, 2 + 2− 1 = 2, 3π(p2) = (2, 4)π(p3) = (2, 4)L = {(1, 3), (3, 4)}b(e) = 4− 2 = 2b(e)− x(e) + r(pi) = 2− 3 + 2 = 1 < r(pi) = 2We recalculate x

Table 3. Recalculation of x in Algorithm 4, second iteration

arc (1, 2) (1, 3) (2, 1) (2, 4) (2, 3) (3, 1) (3, 2) (3, 4) (4, 2) (4, 3)b 4 3 0 2 1 3 1 3 3 1x 3 2 0 2 1 0 0 3 0 0b(l)− x(l)|l ∈ L 1 0

T = 6.4i = 2 + 2 = 4 = q

Third iteration:e = max{b(e)− x(e)|e ∈ L} = (1, 3)j = 1i+ j − 1 = 4 + 1− 1 = 4π(p4) = (1, 3)


Figure 2. A randomly generated weakly connected directedgraph with n = 50 and edge density 20%

L = {(3, 4)}b(e) = 3− 1 = 2b(e)− x(e) + r(pi) = 2− 2 + 1 = 1 ≮ r(pi) = r(p4) = 1i = 4 + 1 = 5 > qSince i > q, the algorithm terminates and the solution is: π(f1) = π(p3) =

(2, 4), π(f2) = π(p1) = (2, 1), π(f3) = π(p2) = (2, 4), π(f4) = π(p4) = (1, 3). Thestatic flow corresponding to the quickest flow x is given in the following table withthe quickest time T = 6.4.

arc (1, 2) (1, 3) (2, 1) (2, 4) (2, 3) (3, 1) (3,2) (3, 4) (4, 2) (4, 3)b 4 2 0 2 1 3 1 3 3 1x 3 2 0 2 1 0 0 3 0 0

The quickest time T calculated by MILP solver for this example is 6.2 which isvery close to result obtained by the heuristic. Worth noting, in this example, is thatif e = (1, 3) is chosen in the second iteration, the result of the heuristic coincideswith that of the MILP solver.

3.3. Computational experiment. For the computations, we have generated ran-dom weakly connected directed graphs with n nodes and 20% edge density. Onesuch graph with n = 50 is shown in Figure 2. The capacity of each arc is taken ran-domly between 1 to 5 units of flow (e.g. cars) per second, and the travel time on eacharc between 1 minute to 10 minutes. A random sample of 20% of the arcs is chosenas L, and 1 ≤ ν(l) ≤ 5,∀l ∈ L. Taking F = 10, 000, with n ∈ {50, 100, 200, 400}


and |P | ∈ {50, 75, 100}, we have taken the average running time on 10 graphs perinstance.

The running times of Algorithm 4 with the mixed integer programming formula-tion are recorded in Table 4. For an instance of n = 400, |P | = 50, the MILP solvertook more than 45 minutes of running time and we have not recorded the time inthe table. For an instance of n = 1000, |P | = 50, Algorithm 4 took 37 seconds andCBC solver was unable to find the solution.

Table 4. Running time (in seconds) of Algorithm 4 and mixed in-teger programming formulation of multi-facility quickest FlowLocproblem

n |P | = 50 |P | = 75 |P | = 100

50Algorithm 4 0.32 0.23 0.22

MILP 4.46 9.05 14.27

100Algorithm 4 0.49 0.48 0.45

MILP 27.76 50.71 77.88

200Algorithm 4 1.48 1.45 1.46

MILP 232.47 463.96 714.74

400Algorithm 4 5.44 5.46 5.40

MILP - - -

In each instance, the percentage deviation of the objective function value fromthe MILP objective function value is found to be zero, i.e. all the objective functionvalues match in the random experiments. It means that the proposed algorithmworks well in practice.

For the implementation of algorithms and solution of mixed integer program-ming, we have used Python 3.7 on a computer with Intel® Core™ i5, 2.30 GHzprocessor, 4GB RAM, and 64-bit operating system. The mixed integer programhas been solved using CBC (Coin-OR branch and cut) solver.

4. Quickest FlowLoc problem with contraflow

Assuming that the direction of arcs in a directed network can be reversed (i.e. thedirection of the traffic flow on a road segment can be reversed), a contraflow problemseeks to choose the ideal direction of arcs to optimize network flows. To solvea dynamic contraflow problem on a network N = (V,A, b, τ, s, d), an undirectednetwork N = (V, A, b, τ , s, d), known as auxiliary network of N , is constructed suchthat

A = {(u,w) : (u,w) ∈ A or (w, u) ∈ A}For each (u,w) ∈ A,

b(u,w) = b(u,w) + b(w, u)

τ(u,w) =

{τ(u,w) if (u,w) ∈ Aτ(w, u) otherwise

in which we consider b(i, j) = 0 whenever (i, j) /∈ A, and vice versa.Implementation of algorithms to solve various dynamic flow problems on aux-

iliary network helps to solve the corresponding contraflow problems [32, 24]. Forexample, to solve the maximum contraflow problem which seeks to maximize the


1

2

3

4

7,17,1

7,37,3

6,46,4

4,14,1

2,12,1

Figure 3. Auxiliary network N of network N in Figure 1 witharc labels (capacity, travel time)

flow allowing arc reversals in a given network, we solve the maximum flow problemin its auxiliary network. Then, the flow is decomposed into paths and cycles andcycle flows are removed. The analogous procedure to solve the quickest contraflowproblem can be found in Pyakurel et al. [28]. The algorithms to solve the con-traflow problems not only give optimal flow decisions but also an output which arcsto reverse and which arcs not to reverse. In what follows, the flow in the auxiliarynetwork N of N without cycle flows will be referred to as contraflow in N .

If we allow reversal of the direction of the usual traffic flow, especially in case ofemergency evacuation planning, there may be a significant reduction in the quickesttime. The change in the capacity of arcs, in such cases, have also effects on locationdecisions.

Definition 4.1 (Maximum static (dynamic) ContraFlowLoc). Let an evacuationnetwork N = (V,A, b, τ, s, d) be a network with the set of all feasible locationsL ⊆ A, set of all facilities P , the size of the facilities r : P → N and the numberof facilities that can be placed on the possible locations ν : L→ N. The maximumstatic (dynamic) ContraFlowLoc problem asks for an allocation π : P → L, suchthat the static (dynamic) maximum flow value is maximized after the facility-allocation, allowing arc reversals.

Definition 4.2 (Quickest ContraFlowLoc). Given an evacuation network N =(V,A, b, τ, s, d), supply F at s, set of feasible locations L ⊆ A, the set of all facilitiesP , the size of the facilities r : P → N, the number of facilities that can be placed onthe possible locations and ν : L → N, the quickest ContraFlowLoc problem seeksfor an allocation π : P → L of the facilities to the edges, such that the quickesttime to transport F from s to d is minimized, allowing arc reversals.

Example 4. Consider the network given in Figure 1. The labels on arcs de-note capacity and travel time respectively. Let the set of feasible locations L ={(2, 1), (1, 3)} and the size of a single facility r = 2. If the facility is placed on(2, 1), the values of the static maximum flow before and after contraflow configu-ration are 7 (4 along 1− 2− 4 and 3 along 1− 3− 4) and 11 (5 along 1− 2− 4, 4along 1− 3− 4, 2 along 1− 3− 2− 4) respectively. If the facility is placed on (1, 3),the corresponding values are 5 (4 along 1−2− 4, 1 along 1−3− 4) and 11 (7 along


1− 2− 4, 4 along 1-3-4) respectively. Thus the static FlowLoc decision before con-traflow configuration is (2, 1) and after contraflow configuration is (2, 1) or (1, 3).The decisions with the quickest time before and after contraflow configuration withF = 109 are listed in Table 5.

Table 5. Quickest time calculations (cf. Example 4)

Quickest time, F = 109Facility placed on Before contraflow After contraflow

(2, 1) 20 15(1, 3) 25.8 14.27

Location Decision (2, 1) (1, 3)

To solve the single-facility maximum static(dynamic) FlowLoc problem, we caniteratively choose an arc from L, place the facility, reduce its capacity by the size ofthe facility, calculate the maximum static (dynamic) contraflow value, and choosethe arc in which the difference of the maximum contraflow value after placing thefacility and without placing the facility on any arc is the least (see also [6]). Here, wepresent Algorithm 5 to solve the single facility quickest ContraFlowLoc problem,which iteratively chooses an arc from L, reduces its capacity by the size of thefacility r, finds the quickest contraflow and retains its capacity before choosing thenext arc. The arc which gives the minimum quickest time after placing the facilityon it is chosen as the optimal location.

Algorithm 5: Single facility quickest ContraFlowLoc I

Input : Directed network N = (V,A, b, τ, s, d), the set of possible locationsL, supply at s = F , size r of the facility

Output: Location loc of the facility, static contraflow x corresponding tothe quickest contraflow, corresponding quickest time T , set of arcsto be reversed R

1 T =∞2 for l ∈ L do3 b(l) = b(l)− r4 new quickest time = quickest contraflow time in the modified network

if new quickest time < T then5 T = new quickest time

6 loc = l

7 x = the corresponding static contraflow

8 end

9 b(l) = b(l) + r

10 end

11 R = {(j, i) ∈ A|x(i, j) > b(i, j) if (i, j) ∈ A or x(i, j) > 0 if (i, j) /∈ A}12 return loc, x, T,R

We can improve the practical running time of Algorithm 5 by adapting Algorithm3 to the contraflow case. After contraflow calculation, if arc (u, v) ∈ L and its


opposite arc (v, u) ∈ A together have capacity enough to host the facility, then(u, v) is chosen to locate the facility and we can get rid of the remaining |L| − 1quickest contraflow calculations of Algorithm 5. The procedure is elucidated inAlgorithm 6.

Algorithm 6: Single facility quickest ContraFlowLoc II

Input : Directed network N = (V,A, b, τ, s, d), the set of possible locationsL, supply at s = F , size r of the facility

Output: Location loc of the facility, static contraflow x corresponding tothe quickest contraflow, corresponding quickest time T , set of arcsto be reversed R

1 x = static contraflow corresponding to the quickest contraflow N


3 (u∗, v∗) = arg max{b(u, v) + b(v, u)− x(u, v)− x(v, u)|(u, v) ∈ L}4 if b(u∗, v∗) + b(v∗, u∗)− x(u∗, v∗)− x(v∗, u∗) ≥ r then5 loc = (u∗, v∗)

6 b(u∗, v∗) = b(u∗, v∗)− r7 else8 Algorithm 5

9 end

10 R = {(v, u) ∈ A|x(u, v) > b(u, v) if (u, v) ∈ A or x(u, v) > 0 if (u, v) /∈ A}11 return loc, x, T,R

For finding the static contraflow corresponding to the quickest contraflow, wesolve the quickest flow problem in the auxiliary network and remove cycle flows (ifany) (Pyakurel et al. [28]). Because the size of the facility does not exceed thecapacity of an arc in L (Remark 1), and placing the facility on an arc (u, v) ∈ Lreduces the capacity of (u, v) and (v, u) both in the auxiliary network, and removalof the cycle flows in contraflow calculation, Algorithm 5 and Algorithm 6 find thelocation loc with the minimum quickest time. Moreover, the removal cycle flows ina contraflow computation leads either x(u, v) or x(v, u) to vanish so that the set ofarcs to be reversed R is well defined. The discussion leads to:

Lemma 4.3. Algorithm 5 or Algorithm 6 solves the single facility quickest Con-traFlowLoc problem optimally.

Theorem 4.4. The single facility quickest ContraFlowLoc problem can be solvedin strongly polynomial time.

Proof. The complexity of the for loop in Algorithm 5 is dominated by the com-plexity of the quickest flow calculation which can be done in strongly polynomialtime O(nm2 log2 n). Since the auxiliary network can be formed in linear time, flowdecomposition can be done in O(nm) time (Ahuja et al. [1]), the overall complexityof Algorithm 5 is O(|L|nm2 log2 n). �

Since the multi-facility FlowLoc problems are NP -hard, the corresponding Con-traFlowLoc problems are also NP -hard. Replacing quickest flow calculations bymaximum static(dynamic) contraflow calculations and adjusting capacities accord-ingly, Algorithm 4, can also be adapted to construct a polynomial time heuristic


to solve the corresponding multi-facility case. To solve the multi-facility quickestContraFlowLoc problem, we present such an adaptation in Algorithm 7.

Algorithm 7: Multi facility quickest ContraFlowLoc heuristic

Input : Directed network N = (V,A, b, τ, s, d), the set of possible locationsL, supply F at the source s, set of facilities P with size r : P → N

Output: Allocation π : P → L, the quickest time T after allocation, set ofarcs to be reversed R

1 sort facilities according to their size r(p1) ≥ r(p2) ≥ · · · ≥ r(pq)2 x = static contra flow corresponding to the quickest contra flow

3 d(u, v) = b(u, v) + b(v, u)− x(u, v)− x(v, u) ∀(u, v) ∈ A4 T = the corresponding quickest time

5 i = 1

6 while i ≤ q do7 (u∗, v∗) = arg max{d(u, v)|(u, v) ∈ L}8 for j = 1 to j = ν(u∗, v∗) do9 if i+ j − 1 ≤ q then

10 π(pi+j−1) = (u∗, v∗)

11 end

12 end

13 L = L \ {(u∗, v∗)}14 b(u∗, v∗) = b(u∗, v∗)− r(pi)15 if d(u∗, v∗) + r(pi) < r(pi) then16 x = static contra flow corresponding to the quickest contra flow with

modified b17 T = the corresponding quickest time

18 end

19 i = i+ ν(u∗, v∗)

20 end

21 R = {(v, u) ∈ A|x(u, v) > b(u, v) if (u, v) ∈ A or x(u, v) > 0 if (u, v) /∈ A}22 return π, loc, T,R

Example 5. To illustrate Algorithm 7, we reconsider the problem illustrated inExample 3 with the possibility of arc reversals. The static contra flow correspondingto the quickest contra flow are tabulated in the following table.

arc (1, 2) (1, 3) (2, 1) (2, 4) (2, 3) (3, 1) (3, 2) (3, 4) (4, 2) (4, 3)b 4 3 3 4 1 3 1 3 3 1x 7 2 0 5 2 0 0 4 0 0d(u, v)|(u, v) ∈ L 1 3 1 0

T = 5.22q = 4i = 1 < q

First iteration:(u∗, v∗) = max{d(u, v)|(u, v) ∈ L} = (1, 3)


j = 1i+ j − 1 = 1 + 1− 1 = 1π(p1) = (1, 3)L = {(2, 1), (2, 4), (3, 4)}b(u∗, v∗) = 3− 3 = 0d(u∗, v∗) + r(pi) = 0 + 3− 2− 0 + 3 = 4 ≮ r(p1) = 3i = 1 + 1 = 2 < q

Second iteration: (u∗, v∗) = max{d(u, v)|(u, v) ∈ L} = (2, 4)j = 1, 2i+ j − 1 = 2 + 1− 1, 2 + 2− 1 = 2, 3π(p2) = (2, 4)π(p3) = (2, 4)L = {(2, 1), (3, 4)}b(u∗, v∗) = 4− 2 = 2d(u∗, v∗) + r(pi) = 2 + 3− 5− 0 + 2 = 2 ≮ r(p2) = 2i = 2 + 2 = 4 = q

Third iteration:(u∗, v∗) = max{d(u, v)|(u, v) ∈ L} = (2, 1)j = 1i+ j − 1 = 4 + 1− 1 = 4π(p4) = (2, 1)L = {(3, 4)}b(u∗, v∗) = 3− 1 = 2d(u∗, v∗) + r(pi) = 2 + 4− 7− 0 + 1 = 0 < r(p4) = 1Recalculation of x:

arc (1, 2) (1, 3) (2, 1) (2, 4) (2, 3) (3, 1) (3, 2) (3, 4) (4, 2) (4, 3)b 4 0 2 2 1 3 1 3 3 1x 6 2 0 4 2 0 0 4 0 0

i = 4 + 1 = 5 > q

The solution is: π(f1) = π(p3) = (2, 4), π(f2) = π(p1) = (1, 3), π(f3) = π(p2) =(2, 4), π(f4) = π(p4) = (2, 1). The static contraflow x corresponding to the quickestcontraflow after this allocation is as given in the third iteration. The set of arcsreversed before facility allocation is {(2, 1), (3, 2), (4, 2), (4, 3)} while after allocationit becomes {(2, 1), (3, 2), (4, 2), (4, 3), (1, 3)}. The quickest time is 5.375 which isless than the quickest time 6.2 of the same problem without arc reversals. Thesignificance of the contraflow approach is that the difference between the quickesttimes before and after arc reversals increase with the growing value of F . Someobservations of this problem, after facility-allocation are listed in the following table.

Quickest timeF Before contraflow After contraflow

100 24.2 15.331000 204.2 115.3310000 2004.2 1115.33


5. Conclusion

In an effort to combine location decisions with the network flow models, in thispaper models to combine quickest flow with location analysis are introduced. Witha view to be applied in traffic flow management in emergency evacuation planning,the facility-arc assignments are done so as to affect the quickest time of evacuationthe least. Exact polynomial algorithms to assign single facility and polynomial timeheuristic algorithms to place multiple facilities on multiple arcs (with and withoutthe possibility of arc reversal) are designed. As in the absence of facility allocations,a significant improvement in the quickest time has been achieved with arc reversals,i.e. with contraflow configuration in the FlowLoc case also. Presented algorithmsperform very well in randomly generated graphs, so that they can be used to makefacility location decisions in emergency evacuation problems. The algorithms areparticularly important when a known volume of evacuees from a danger zone hasto be transferred to a safe zone with the least possible interruption in the evacua-tion time because of facility allocation in some road segments of the transportationnetwork. To the best of our knowledge, the problems and the corresponding algo-rithms to solve the quickest FlowLoc problems are considered for the first time inthis paper.

However, in the problems considered, the locations decisions are made on thebasis of quickest flow with a background of maximum flow in a single-source-single-sink network with constant capacity and transit time. Also the number of availablefacilities does not exceed the number of available locations, and the size of eachfacility fits in any of the available locations. Hence, the similar problems with otheraspects of network flow in more generalized settings can be the natural extensionsof the problem.

References

1. R.K. Ahuja, T.L. Magnati, and J.B. Orlin (1988). Network flows, Massachusetts Institute of

Technology, Operations Research Center.2. S. An, N. Cui, X. Li, and Y. Ouyang (2013). Location planning for transit-based evacuation

under the risk of service disruptions. Transportation Research Part B: Methodological, 54,

1-16.3. R.D. Armstrong and Z. Jin (1997). A new strongly polynomial dual network simplex algorithm.

Mathematical programming, 78(2), 131-148.

4. T.N. Dhamala and U. Pyakurel (2013). Earliest arrival contraflow problem on series-parallelgraphs. International Journal of Operations Research, 10, 1-13

5. T.N. Dhamala, U. Pyakurel and S. Dempe (2018). A critical survey onthe network optimization algorithms for evacuation planning problems.

International Journal of Operations Research, 15(3), 101-133.

6. R.C. Dhungana and T.N. Dhamala (2018). FlowLoc Problems on Evacuation Network. InThe 11th Triennial Conference of Association of Asia Pacific Operational Research Societies

(APORS 2018, August 6-9), 121-123.

7. R.C. Dhungana, U. Pyakurel and T.N. Dhamala (2018). Abstract contraflow models and so-lution procedures for evacuation planning. Journal of Mathematics Research, 10(4), 89-100.

8. F.R. Ford and D.R. Fulkerson (1958). Constructing maximal dynamic flows from static flows.

Operations Research, 6, 419-433.9. M. Goerigk, K. Deghdak, and P. Heßler (2014). A comprehensive evacuation planning model

and genetic soution algorithm. Transportation Research, Part E, 71, 82-97

10. M. Goerigk, B. Grun and P. Heßler (2014). Combining bus evacuation with location decisions:A branch-and-price approach, Transportation Research Procedia, 2, 783-791.


11. H.W. Hamacher, S. Heller, and B. Rupp (2013), Flow location (FlowLoc) problems: dynamic

network flows and location models for evacuation planning. Annals of Operations Research,

207(1), 161–180.12. S. Heller and H.W. Hamacher (2011). The multi-terminal q-FlowLoc problem: a heuristic,

In Lecture Notes in Computer Science, 6701, Proceedings of the International Network Opti-

mization Conference, 523-528, Berlin, Springer.13. H. Jia, F. Ordonez and M. Dessouky (2007). A modeling framework for facility location of

medical services for large-scale emergencies. IIE transactions, 39(1), 41-55.

14. S. Kim, S. Shekhar and M. Min (2008). Contraflow transportation network reconfigurationfor evacuation route planning. IEEE Transactions on Knowledge and Data Engineering, 20,

1-15.

15. S. Kongsomsaksakul, C. Yang, and A. Chen (2005). Shelter location-allocation model forflood evacuation planning. Journal of the Eastern Asia Society for Transportation Studies, 6,

4237-4252.16. E. Kohler, K. Langkau and M. Skutella (2002). Time expanded graphs for flow depended

transit times. R. Mohring and R. Raman (Eds.): ESA 2002, LNCS 2461, Springer-Verlag,

599-611.17. A. Kulshrestha, Y. Lou, and Y. Yin (2014). Pickup locations and bus allocation for tran-

sitbased evacuation planning with demand uncertainty. Journal of Advanced Transportation,

48(7), 721-733.18. M. Lin and P. Jaillet (2015). On the quickest flow problem in dynamic networks− a parametric

min-cost flow approach. Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on

Discrete Algorithms, 1343-1356.19. M. Ng, J. Park, and S.T. Waller (2010). A hybrid bilevel model for the optimal shelter

assignment in emergency evacuations. ComputerAided Civil and Infrastructure Engineering,

25(8), 547-556.20. J. B. Orlin (2013). Max flows in O(nm) time, or better. In Proceedings of the forty-fifth annual

ACM symposium on Theory of computing, 765-774.21. U. Pyakurel (2016). Evacuation planning problem with contraflow approach. PhD Thesis,

IOST, Tribhuvan University, Nepal.

22. U. Pyakurel and T.N. Dhamala (2014). Earliest arrival contraflow model for evacuation plan-ning. N eural, Parallel, and Scientific Computations [CNLS-2013], 22, 287-294.

23. U. Pyakurel and T.N. Dhamala (2015). Models and algorithms on contraflow evacuation plan-

ning network problems. I nternational Journal of Operations Research, 12, 36-46.24. U. Pyakurel and T.N. Dhamala (2016). Continuous time dynamic contraflow models and

algorithms. Advances in Operations Research - Hindawi ; Article ID 368587, 1-7.

25. U. Pyakurel and T.N. Dhamala (2017a). Evacuation planning by earliest arrival contraflow,Journal of Industrial and Management Optimization 13, 489-503.

26. U. Pyakurel and T.N. Dhamala (2017b). Continuous dynamic contraflow approach for evacu-

ation planning. Annals of Operation Research, 253, 573-598.27. U. Pyakurel, T.N. Dhamala and S. Dempe (2017a). Efficient continuous contraflow algorithms

for evacuation planning problems. Annals of Operations Research (ANOR), 254, 335-364.28. U. Pyakurel, H.N. Nath and T.N. Dhamala (2018). Efficient contraflow algorithms for quickest

evacuation planning. Science China Mathematics, 61(11), 2079-2100.

29. U. Pyakurel, H.H. Nath and T.N. Dhamala (2018). Partial contraflow with path reversals forevacuation planning. Annals of Operations Research, DOI: 10.1007/s10479-018-3031-8.

30. U. Pyakurel, S. Dempe and T.N. Dhamala (2018). Efficient algorithms for flow over time

evacuation planning problems with lane reversal strategy. TU Bergakademie Freiberg.31. U. Pyakurel, H.W. Hamacher and T.N. Dhamala (2014). Generalized maximum dynamic

contraflow on lossy network. International Journal of Operations Research Nepal, 3, 27-44.

32. S. Rebennack, A. Arulselvan, L. Elefteriadou and P.M. Pardalos (2010). Complexity analysisfor maximum flow problems with arc reversals. Journal of Combinatorial Optimization, 19,

200-216.

33. B. Rupp (2010). FlowLoc: Discrete facility locations in flow networks, Diploma thesis, Uni-versity of Kaiserslautern, Germany.

34. S. Ruzika, H. Sperber and M. Steiner (2011). Earliest arrival flows on seriesparallel graphs.

Networks, 57(2), 169-173.


35. M. Saho and M. Shigeno (2017). Cancel-and-tighten algorithm for quickest flow problems.

Network, 69(2), 179-188.

36. H.D. Sherali, T.B. Carter, and A.G. Hobeika (1991). A location-allocation model and algo-rithm for evacuation planning under hurricane/flood conditions. Transportation Research Part

B: Methodological, 25(6), 439-452.

37. Y. Sheffi (1985). Urban transportation networks: Equilibrium analysis with mathematicalprogramming methods. Prentice-Hall, Englewood Cliffs.

38. M. Skutella (2009). An introduction to network flows over time. In Research trends in com-

binatorial optimization, 451-482.39. Torres, F. E. (1990). Linearization of mixed-integer products. Mathematical programming,

49(1), 427-428.

1Tribhuvan University, Bhaktpur Multiple Campus, Bhaktpur, Nepal, 2,3 CentralDepartment of Mathematics, Tribhuvan University, P.O.Box 13143, Kathmandu, Nepal;4 TU Bergakademie, Fakultat fur Mathematik und Informatik, 09596 Freiberg, Ger-

many; 2 Currently: TU Bergakademie, Fakultat fur Mathematik und Informatik, 09596Freiberg, Germany

E-mail address: [email protected], [email protected]

E-mail address: [email protected], [email protected]

Documents

H. N. Nath, U. Pyakurel, T. N. Dhamala, and S. Dempe ... · in pseudo-polynomial time. They also introduce the lex-maximum dynamic con-tra ow problem in which ow is maximized in given